West Coast. But ride from North to South... less windy. There are great books on the route plus maps by Adventure Cycling. Lots of cyclists do this one so there are usually a few other bikers at the campsites. Most camping areas have hikker/bikker sites so you don't camp next to a RN and they're usually $5-15 a night. Many of the campsites have laundry on site too.
The route is pretty good for beginners. There are some narrower sections of road but usually not to narrow, and the speed limit is slower for most of the route. The weather is predictable and pleasant, the topography is mostly flat with little rolling hills.
Vancouver to San Fransisco is a good starter. Give it 2-3 weeks.